Great question — answering as the UphillTech team:
Assuming it isn't condensation, one thing that causes eggs to weep is the breakdown of the protein structures. When you cook an egg, the heat causes the proteins to unfold, then refold into different shapes, capturing liquid in the process which is what allows the boiled egg to hold it's structure. These bonds are fairly weak and can be broken down with salt and acids--two things which you have in your deviled eggs. One thing that might help is slowing down the unfolding/refolding process by putting the eggs in before the water is fully heated and not heating the water up to a full boil. By slowing down the process you are building stronger bonds that won't break as easily. You might also...
Full answer by Mark Burnett: https://cooking.stackexchange.com/a/84955 (CC BY-SA 3.0).
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